8. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant.
The point \(P \left( a p ^ { 2 } , 2 a p \right)\) lies on the parabola \(C\).
- Show that an equation of the tangent to \(C\) at \(P\) is
$$p y = x + a p ^ { 2 }$$
The tangent to \(C\) at the point \(P\) intersects the directrix of \(C\) at the point \(B\) and intersects the \(x\)-axis at the point \(D\).
Given that the \(y\)-coordinate of \(B\) is \(\frac { 5 } { 6 } a\) and \(p > 0\),
- find, in terms of \(a\), the \(x\)-coordinate of \(D\).
Given that \(O\) is the origin,
- find, in terms of \(a\), the area of the triangle \(O P D\), giving your answer in its simplest form.